Numerical Differentiation - Examples & Applications

Numerical Differentiation - Examples & Applications

This section provides practical examples of numerical differentiation techniques, including forward, backward, and centered finite differences, high-accuracy formulas, Richardson extrapolation, unequally spaced data, and partial derivatives.

First-Derivative Finite Differences

The most basic methods for estimating the first derivative are the forward, backward, and centered divided difference formulas, all derived from the Taylor series expansion.

Intermediate: Comparing Forward, Backward, and Centered Differences

Estimate the first derivative of

f(x) = -0.1x^4 - 0.15x^3 - 0.5x^2 - 0.25x + 1.2

x = 0.5

using a step size of

h = 0.25

. Calculate the true percent relative error for the forward, backward, and centered difference approximations. The true value of the derivative is

f'(0.5) = -0.9125

Step-by-Step Solution

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Higher-Accuracy Differentiation Formulas

By retaining more terms in the Taylor series expansion, we can derive finite difference formulas with higher-order accuracy (e.g.,

O(h^2)

for forward/backward, and

O(h^4)

for centered).

Advanced: High-Accuracy Centered Difference

Estimate the first derivative of

f(x) = -0.1x^4 - 0.15x^3 - 0.5x^2 - 0.25x + 1.2

x = 0.5

using the

O(h^4)

centered difference formula. Use a step size of

h = 0.25

. The true value is

-0.9125

Step-by-Step Solution

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Derivatives of Unequally Spaced Data

When data points are not equally spaced, standard finite difference formulas cannot be used. Instead, a Lagrange interpolating polynomial is fit through the points and then differentiated.

Intermediate: Unequally Spaced Data

Given three unequally spaced points:

x_0 = 1, f(x_0) = 0

x_1 = 4, f(x_1) = 1.386

x_2 = 6, f(x_2) = 1.792

Estimate the derivative

f'(4)

Step-by-Step Solution

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Partial Derivatives

Partial derivatives are estimated by holding all other variables constant and applying standard finite difference formulas to the variable of interest.

Intermediate: Numerical Partial Derivative

Let

f(x, y) = x^2 y + 3xy^2

. Estimate

\frac{\partial f}{\partial x}

(1, 2)

using a centered difference with

h_x = 0.1

Step-by-Step Solution

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Richardson Extrapolation

Richardson extrapolation uses two derivative estimates computed with different step sizes to obtain a third, more accurate estimate.

Advanced: Richardson Extrapolation

Use Richardson extrapolation to estimate the first derivative of

f(x) = \ln(x)

x = 2

. First, calculate centered difference estimates using

h_1 = 0.2

and

h_2 = 0.1

. Then, combine them using Richardson extrapolation to find an

O(h^4)

estimate. True value is

f'(2) = 1/2 = 0.5

Step-by-Step Solution

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